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Numerical evaluation of the Kummer function with complex argument by the trapezoidal rule. (English) Zbl 0779.65013
The Kummer function \(\Phi(a,c;z)\), expressed in its well-known integral form, is numerically evaluated for complex values of \(z\) by means of the trapezoidal rule. The achieved high degree of accuracy is explained through a detailed investigation of the related Euler-Maclaurin formula. Theoretically interesting error bounds are given. Recurrence relations are used for obtaining \(a\) and \(c\) values for which the trapezoidal rule is not suitable. The procedure is compared with one based on Gauss-Jacobi quadrature.
65D20 Computation of special functions and constants, construction of tables
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)